/* @(#)s_erf.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
   for performance improvement on pipelined processors.
*/

/* double erf(double x)
 * double erfc(double x)
 *			     x
 *		      2      |\
 *     erf(x)  =  ---------  | exp(-t*t)dt
 *	 	   sqrt(pi) \|
 *			     0
 *
 *     erfc(x) =  1-erf(x)
 *  Note that
 *		erf(-x) = -erf(x)
 *		erfc(-x) = 2 - erfc(x)
 *
 * Method:
 *	1. For |x| in [0, 0.84375]
 *	    erf(x)  = x + x*R(x^2)
 *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
 *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
 *	   where R = P/Q where P is an odd poly of degree 8 and
 *	   Q is an odd poly of degree 10.
 *						 -57.90
 *			| R - (erf(x)-x)/x | <= 2
 *
 *
 *	   Remark. The formula is derived by noting
 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
 *	   and that
 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
 *	   is close to one. The interval is chosen because the fix
 *	   point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
 *	   near 0.6174), and by some experiment, 0.84375 is chosen to
 * 	   guarantee the error is less than one ulp for erf.
 *
 *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
 *         c = 0.84506291151 rounded to single (24 bits)
 *         	erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
 *         	erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
 *			  1+(c+P1(s)/Q1(s))    if x < 0
 *         	|P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
 *	   Remark: here we use the taylor series expansion at x=1.
 *		erf(1+s) = erf(1) + s*Poly(s)
 *			 = 0.845.. + P1(s)/Q1(s)
 *	   That is, we use rational approximation to approximate
 *			erf(1+s) - (c = (single)0.84506291151)
 *	   Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
 *	   where
 *		P1(s) = degree 6 poly in s
 *		Q1(s) = degree 6 poly in s
 *
 *      3. For x in [1.25,1/0.35(~2.857143)],
 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
 *         	erf(x)  = 1 - erfc(x)
 *	   where
 *		R1(z) = degree 7 poly in z, (z=1/x^2)
 *		S1(z) = degree 8 poly in z
 *
 *      4. For x in [1/0.35,28]
 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
 *			= 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
 *			= 2.0 - tiny		(if x <= -6)
 *         	erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
 *         	erf(x)  = sign(x)*(1.0 - tiny)
 *	   where
 *		R2(z) = degree 6 poly in z, (z=1/x^2)
 *		S2(z) = degree 7 poly in z
 *
 *      Note1:
 *	   To compute exp(-x*x-0.5625+R/S), let s be a single
 *	   precision number and s := x; then
 *		-x*x = -s*s + (s-x)*(s+x)
 *	        exp(-x*x-0.5626+R/S) =
 *			exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
 *      Note2:
 *	   Here 4 and 5 make use of the asymptotic series
 *			  exp(-x*x)
 *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
 *			  x*sqrt(pi)
 *	   We use rational approximation to approximate
 *      	g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
 *	   Here is the error bound for R1/S1 and R2/S2
 *      	|R1/S1 - f(x)|  < 2**(-62.57)
 *      	|R2/S2 - f(x)|  < 2**(-61.52)
 *
 *      5. For inf > x >= 28
 *         	erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
 *         	erfc(x) = tiny*tiny (raise underflow) if x > 0
 *			= 2 - tiny if x<0
 *
 *      7. Special case:
 *         	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 *         	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
 *	   	erfc/erf(NaN) is NaN
 */

#include <math.h>
typedef int int32_t;
typedef unsigned int u_int32_t;

#define __ieee754_exp __mth_i_dexp
extern double __ieee754_exp(double);

/*  LITTLE ENDIAN IEEE format  */
typedef union {
  double value;
  struct {
    u_int32_t lsw;
    u_int32_t msw;
  } parts;
} ieee_double_shape_type;

/* Get the more significant 32 bit int from a double.  */

#define GET_HIGH_WORD(i, d)                                                    \
  do {                                                                         \
    ieee_double_shape_type gh_u;                                               \
    gh_u.value = (d);                                                          \
    (i) = gh_u.parts.msw;                                                      \
  } while (0)

/* Set the less significant 32 bits of a double from an int.  */

#define SET_LOW_WORD(d, v)                                                     \
  do {                                                                         \
    ieee_double_shape_type sl_u;                                               \
    sl_u.value = (d);                                                          \
    sl_u.parts.lsw = (v);                                                      \
    (d) = sl_u.value;                                                          \
  } while (0)

static const double tiny = 1e-300,
                    half =
                        5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
    one = 1.00000000000000000000e+00,               /* 0x3FF00000, 0x00000000 */
    two = 2.00000000000000000000e+00,               /* 0x40000000, 0x00000000 */
    /* c = (float)0.84506291151 */
    erx = 8.45062911510467529297e-01,     /* 0x3FEB0AC1, 0x60000000 */
                                          /*
                                           * Coefficients for approximation to  erf on [0,0.84375]
                                           */
    efx = 1.28379167095512586316e-01,     /* 0x3FC06EBA, 0x8214DB69 */
    efx8 = 1.02703333676410069053e+00,    /* 0x3FF06EBA, 0x8214DB69 */
    pp[] = {1.28379167095512558561e-01,   /* 0x3FC06EBA, 0x8214DB68 */
            -3.25042107247001499370e-01,  /* 0xBFD4CD7D, 0x691CB913 */
            -2.84817495755985104766e-02,  /* 0xBF9D2A51, 0xDBD7194F */
            -5.77027029648944159157e-03,  /* 0xBF77A291, 0x236668E4 */
            -2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */
    qq[] = {0.0,
            3.97917223959155352819e-01,   /* 0x3FD97779, 0xCDDADC09 */
            6.50222499887672944485e-02,   /* 0x3FB0A54C, 0x5536CEBA */
            5.08130628187576562776e-03,   /* 0x3F74D022, 0xC4D36B0F */
            1.32494738004321644526e-04,   /* 0x3F215DC9, 0x221C1A10 */
            -3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */
                                          /*
                                           * Coefficients for approximation to  erf  in [0.84375,1.25]
                                           */
    pa[] = {-2.36211856075265944077e-03,  /* 0xBF6359B8, 0xBEF77538 */
            4.14856118683748331666e-01,   /* 0x3FDA8D00, 0xAD92B34D */
            -3.72207876035701323847e-01,  /* 0xBFD7D240, 0xFBB8C3F1 */
            3.18346619901161753674e-01,   /* 0x3FD45FCA, 0x805120E4 */
            -1.10894694282396677476e-01,  /* 0xBFBC6398, 0x3D3E28EC */
            3.54783043256182359371e-02,   /* 0x3FA22A36, 0x599795EB */
            -2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */
    qa[] = {0.0,
            1.06420880400844228286e-01,   /* 0x3FBB3E66, 0x18EEE323 */
            5.40397917702171048937e-01,   /* 0x3FE14AF0, 0x92EB6F33 */
            7.18286544141962662868e-02,   /* 0x3FB2635C, 0xD99FE9A7 */
            1.26171219808761642112e-01,   /* 0x3FC02660, 0xE763351F */
            1.36370839120290507362e-02,   /* 0x3F8BEDC2, 0x6B51DD1C */
            1.19844998467991074170e-02},  /* 0x3F888B54, 0x5735151D */
                                          /*
                                           * Coefficients for approximation to  erfc in [1.25,1/0.35]
                                           */
    ra[] = {-9.86494403484714822705e-03,  /* 0xBF843412, 0x600D6435 */
            -6.93858572707181764372e-01,  /* 0xBFE63416, 0xE4BA7360 */
            -1.05586262253232909814e+01,  /* 0xC0251E04, 0x41B0E726 */
            -6.23753324503260060396e+01,  /* 0xC04F300A, 0xE4CBA38D */
            -1.62396669462573470355e+02,  /* 0xC0644CB1, 0x84282266 */
            -1.84605092906711035994e+02,  /* 0xC067135C, 0xEBCCABB2 */
            -8.12874355063065934246e+01,  /* 0xC0545265, 0x57E4D2F2 */
            -9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */
    sa[] = {0.0,
            1.96512716674392571292e+01,   /* 0x4033A6B9, 0xBD707687 */
            1.37657754143519042600e+02,   /* 0x4061350C, 0x526AE721 */
            4.34565877475229228821e+02,   /* 0x407B290D, 0xD58A1A71 */
            6.45387271733267880336e+02,   /* 0x40842B19, 0x21EC2868 */
            4.29008140027567833386e+02,   /* 0x407AD021, 0x57700314 */
            1.08635005541779435134e+02,   /* 0x405B28A3, 0xEE48AE2C */
            6.57024977031928170135e+00,   /* 0x401A47EF, 0x8E484A93 */
            -6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */
                                          /*
                                           * Coefficients for approximation to  erfc in [1/.35,28]
                                           */
    rb[] = {-9.86494292470009928597e-03,  /* 0xBF843412, 0x39E86F4A */
            -7.99283237680523006574e-01,  /* 0xBFE993BA, 0x70C285DE */
            -1.77579549177547519889e+01,  /* 0xC031C209, 0x555F995A */
            -1.60636384855821916062e+02,  /* 0xC064145D, 0x43C5ED98 */
            -6.37566443368389627722e+02,  /* 0xC083EC88, 0x1375F228 */
            -1.02509513161107724954e+03,  /* 0xC0900461, 0x6A2E5992 */
            -4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */
    sb[] = {0.0,
            3.03380607434824582924e+01,   /* 0x403E568B, 0x261D5190 */
            3.25792512996573918826e+02,   /* 0x40745CAE, 0x221B9F0A */
            1.53672958608443695994e+03,   /* 0x409802EB, 0x189D5118 */
            3.19985821950859553908e+03,   /* 0x40A8FFB7, 0x688C246A */
            2.55305040643316442583e+03,   /* 0x40A3F219, 0xCEDF3BE6 */
            4.74528541206955367215e+02,   /* 0x407DA874, 0xE79FE763 */
            -2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */

double
__erf(double x)
{
  int32_t hx, ix, i;
  double R, S, P, Q, s, y, z, r;
  GET_HIGH_WORD(hx, x);
  ix = hx & 0x7fffffff;
  if (ix >= 0x7ff00000) { /* erf(nan)=nan */
    i = ((u_int32_t)hx >> 31) << 1;
    return (double)(1 - i) + one / x; /* erf(+-inf)=+-1 */
  }

  if (ix < 0x3feb0000) { /* |x|<0.84375 */
    double r1, r2, s1, s2, s3, z2, z4;
    if (ix < 0x3e300000) { /* |x|<2**-28 */
      if (ix < 0x00800000)
        return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */
      return x + efx * x;
    }
    z = x * x;
    r1 = pp[0] + z * pp[1];
    z2 = z * z;
    r2 = pp[2] + z * pp[3];
    z4 = z2 * z2;
    s1 = one + z * qq[1];
    s2 = qq[2] + z * qq[3];
    s3 = qq[4] + z * qq[5];
    r = r1 + z2 * r2 + z4 * pp[4];
    s = s1 + z2 * s2 + z4 * s3;
    y = r / s;
    return x + x * y;
  }
  if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
    double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
    s = fabs(x) - one;
    P1 = pa[0] + s * pa[1];
    s2 = s * s;
    Q1 = one + s * qa[1];
    s4 = s2 * s2;
    P2 = pa[2] + s * pa[3];
    s6 = s4 * s2;
    Q2 = qa[2] + s * qa[3];
    P3 = pa[4] + s * pa[5];
    Q3 = qa[4] + s * qa[5];
    P4 = s6 * pa[6];
    Q4 = s6 * qa[6];
    P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
    Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
    if (hx >= 0)
      return erx + P / Q;
    else
      return -erx - P / Q;
  }
  if (ix >= 0x40180000) { /* inf>|x|>=6 */
    if (hx >= 0)
      return one - tiny;
    else
      return tiny - one;
  }
  x = fabs(x);
  s = one / (x * x);
  if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */
    double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
    R1 = ra[0] + s * ra[1];
    s2 = s * s;
    S1 = one + s * sa[1];
    s4 = s2 * s2;
    R2 = ra[2] + s * ra[3];
    s6 = s4 * s2;
    S2 = sa[2] + s * sa[3];
    s8 = s4 * s4;
    R3 = ra[4] + s * ra[5];
    S3 = sa[4] + s * sa[5];
    R4 = ra[6] + s * ra[7];
    S4 = sa[6] + s * sa[7];
    R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
    S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
  } else { /* |x| >= 1/0.35 */
    double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
    R1 = rb[0] + s * rb[1];
    s2 = s * s;
    S1 = one + s * sb[1];
    s4 = s2 * s2;
    R2 = rb[2] + s * rb[3];
    s6 = s4 * s2;
    S2 = sb[2] + s * sb[3];
    R3 = rb[4] + s * rb[5];
    S3 = sb[4] + s * sb[5];
    S4 = sb[6] + s * sb[7];
    R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
    S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
  }
  z = x;
  SET_LOW_WORD(z, 0);
  r = __ieee754_exp(-z * z - 0.5625) * __ieee754_exp((z - x) * (z + x) + R / S);
  if (hx >= 0)
    return one - r / x;
  else
    return r / x - one;
}

double
__erfc(double x)
{
  int32_t hx, ix;
  double R, S, P, Q, s, y, z, r;
  GET_HIGH_WORD(hx, x);
  ix = hx & 0x7fffffff;
  if (ix >= 0x7ff00000) { /* erfc(nan)=nan */
                          /* erfc(+-inf)=0,2 */
    return (double)(((u_int32_t)hx >> 31) << 1) + one / x;
  }

  if (ix < 0x3feb0000) { /* |x|<0.84375 */
    double r1, r2, s1, s2, s3, z2, z4;
    if (ix < 0x3c700000) /* |x|<2**-56 */
      return one - x;
    z = x * x;
    r1 = pp[0] + z * pp[1];
    z2 = z * z;
    r2 = pp[2] + z * pp[3];
    z4 = z2 * z2;
    s1 = one + z * qq[1];
    s2 = qq[2] + z * qq[3];
    s3 = qq[4] + z * qq[5];
    r = r1 + z2 * r2 + z4 * pp[4];
    s = s1 + z2 * s2 + z4 * s3;
    y = r / s;
    if (hx < 0x3fd00000) { /* x<1/4 */
      return one - (x + x * y);
    } else {
      r = x * y;
      r += (x - half);
      return half - r;
    }
  }
  if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
    double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
    s = fabs(x) - one;
    P1 = pa[0] + s * pa[1];
    s2 = s * s;
    Q1 = one + s * qa[1];
    s4 = s2 * s2;
    P2 = pa[2] + s * pa[3];
    s6 = s4 * s2;
    Q2 = qa[2] + s * qa[3];
    P3 = pa[4] + s * pa[5];
    Q3 = qa[4] + s * qa[5];
    P4 = s6 * pa[6];
    Q4 = s6 * qa[6];
    P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
    Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
    if (hx >= 0) {
      z = one - erx;
      return z - P / Q;
    } else {
      z = erx + P / Q;
      return one + z;
    }
  }
  if (ix < 0x403c0000) { /* |x|<28 */
    x = fabs(x);
    s = one / (x * x);
    if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
      double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
      R1 = ra[0] + s * ra[1];
      s2 = s * s;
      S1 = one + s * sa[1];
      s4 = s2 * s2;
      R2 = ra[2] + s * ra[3];
      s6 = s4 * s2;
      S2 = sa[2] + s * sa[3];
      s8 = s4 * s4;
      R3 = ra[4] + s * ra[5];
      S3 = sa[4] + s * sa[5];
      R4 = ra[6] + s * ra[7];
      S4 = sa[6] + s * sa[7];
      R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
      S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
    } else { /* |x| >= 1/.35 ~ 2.857143 */
      double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
      if (hx < 0 && ix >= 0x40180000)
        return two - tiny; /* x < -6 */
      R1 = rb[0] + s * rb[1];
      s2 = s * s;
      S1 = one + s * sb[1];
      s4 = s2 * s2;
      R2 = rb[2] + s * rb[3];
      s6 = s4 * s2;
      S2 = sb[2] + s * sb[3];
      R3 = rb[4] + s * rb[5];
      S3 = sb[4] + s * sb[5];
      S4 = sb[6] + s * sb[7];
      R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
      S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
    }
    z = x;
    SET_LOW_WORD(z, 0);
    r = __ieee754_exp(-z * z - 0.5625) *
        __ieee754_exp((z - x) * (z + x) + R / S);
    if (hx > 0)
      return r / x;
    else
      return two - r / x;
  } else {
    if (hx > 0)
      return tiny * tiny;
    else
      return two - tiny;
  }
}
